Fixed-point convergence of modular, steady-state heat transfer models coupling multiple scales and phenomena for melt–crystal growth

Abstract

SUMMARYA block Gauss-Seidel iteration procedure is employed to couple two independent heat transfer codes representing different scales and phenomena in a melt-crystal growth system via shared domain boundaries. In general, such strategies are attractive because of their simplicity for solving problems that may be represented by linking, via shared boundary conditions, existing codes using a modular design. However, these approaches often meet with limited success due to convergence difficulties associated with the overall model. The mathematical framework of fixed-point iterations is employed to assess convergence behaviour for steady-state, non-linear heat transfer models. Analytical forms describe the convergence of one-dimensional problems and provide insight to the behaviour of more complicated, two-dimensional models. Certain implementations that are physically reasonable result in algorithms that will never converge, demonstrating that notions based on physical intuition may not be useful for predicting algorithm performance. A mixing of flux and temperature information along the shared model boundary leads to successful iterative strategies under many conditions.

On favorable thermal fields for detached Bridgman growth

“…The analysis employed here represents another step toward a model that more faithfully depicts the details of heat transfer in technically realistic furnaces. We believe that the next advance will effectively couple these two models with their respective strengths using algorithms that are currently under development [27][28][29][30].…”

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

“…If one step of each solver is taken withx ¼ x ðkÞ andỹ ¼ y ðkÞ , one iteration of the well-known Jacobi method is obtained. For the iterations used here and in our prior efforts [5][6][7][8], a block form of the Gauss-Seidel method is obtained by settingỹ ¼ y ðkÞ , solving for x ðkþ1Þ , then settingx ¼ x ðkþ1Þ to solve for y ðkþ1Þ . Hence, the BGS method consists of sequentially solving each model, rewriting Eqs.…”

“…Pandy and co-workers [5,7] found that block Gauss-Seidel iteration performed poorly for this problem, often failing to converge at all. Derby et al [8] showed that under-relaxed BGS iterations were equally ineffective applied to this same problem.…”

“…The model is described in detail elsewhere [5,7]; we simply repeat the non-dimensionalized equations developed there. Heat transfer occurs by conduction and convection in X 1 and by conduction in X 2 :…”

“…A typical example is the fluid-structure interaction problem, in which the fluid and structure equations are solved separately from one another, subject to boundary conditions that represent self-consistent matching of forces and displacements at the fluid-structure interface [1][2][3][4]. Another example of this type is a conjugate heat transfer problem studied in our earlier work [5][6][7][8], in which a furnace radiation model is coupled to a melt crystal growth model Our method is demonstrated on a problem in melt crystal growth that is partitioned into furnace and growth chamber regions that are modeled by independent heat transfer codes; these are linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss-Seidel iteration fails about half the time for this model problem, quadratic convergence can be achieved by the ABN method under all conditions studied here.…”